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Welcome to a week of Wieferich and Wall-Sun-Sun

This first week of 2012 (1 Jan - 7 Jan) is a home warming party for the new Wieferich and Wall-Sun-Sun prime search projects at PrimeGrid. These projects are looking for extremely rare and unique primes. For more information, please click on the individual project links above. PSA credit is available.

Project stats are available here: Wieferich | Wall-Sun-Sun

They are only available for 64bit OS's (Mac, Linux, Windows). They also require the updated PRPNet 5.0 release. New PRPClient packages are available in the PRPNet thread.

For existing v4.3.7 PRPNet users, it's probably best to get a clean install as there are several updates:

• prpclient v5.0.2 & readme_prpclient.txt
• master-prpclient-ini (new servers and wwww application)
• pfgw v3.6.0 & readme_pfgw.txt
• wwww v1.3 & readme_wwww.txt

For any help or issues with prpclient, please post in the PRPNet Discussion thread.

In order to participate in the two new projects, make sure you give them some project share. Below is an example of 50/50 share:
server=WIEFERICH:50:1:prpnet.mine.nu:13000
server=WALLSUNSUN:50:1:prpnet.mine.nu:13001
Remember that percent across all active projects must equal 100%.

NOTE: As Wall-Sun-Sun is already in first pass territory, please consider setting Wieferich with a higher percent (80/20 for example).

Task lengths are ~1hr for Wieferich and ~45 minutes for Wall-Sun-Sun. If we get a good turnout for this event, it might be best to increase the cache to 2 or more. Below is an example of an 80/20 share with each project requesting 3 WU's at a time:

server=WIEFERICH:80:3:prpnet.mine.nu:13000
server=WALLSUNSUN:20:3:prpnet.mine.nu:13001

stats are still beta and can be found at http://u-g-f.de/PRPNet_beta/

stats of project WFS for Ukraine


stats of project WSS for Ukraine

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анонс на праймгриде

Welcome to the Wieferich Prime Search

A prime p is a Wieferich prime if p^2 divides 2^(p-1) - 1. They are named after Arthur Wieferich who in 1909 proved that if the first case of Fermat’s last theorem is false for the exponent p, then p satisfies the criteria a^(p-1) = 1 (mod p^2) for a=2.

Notice the similarity in the expression p^2 divides 2^(p-1) - 1 to the special case of Fermat's little theorem p divides 2^(p-1) - 1.

Despite a number of extensive searches, the only known Wieferich primes to date are 1093 and 3511. The rarity of these primes has lead to an interest in "Near" Wieferich primes. They are defined as special instances (with small |A|) of 2^((p-1)/2) (mod p^2). Here's a list of "near" Wieferich primes (coming soon) with p > 200.

Search limit Author Year
16000 Beeger 1940
50000 Froberg unknown
100000 Kravitz 1960
200183 Pearson 1964
500000 Riesel 1964
3e7 Froberg 1968
3e9 Brillhart, Tonascia, and Weinberger 1971
6e9 Lehmer 1981
6.1e10 Clark 1996
4e12 Crandall, Dilcher, and Pomerance 1997
4.6e13 Brown and McIntosh 2001
2e14 Crump 2002
1.25e15 Knauer and Richstein 2005
3e15 Carlisle, Crandall, and Rodenkirch 2006
6.7e15 Dorais and Klyve 2011

Above 6.7e15 is unknown at this time. Wieferich@Home is
currently processing the following ranges:

9175868000000000-9177129000000000
9315011000000000-9316478000000000
9505157000000000-9507186000000000
9835792000000000-9837157000000000

Although the search for Wieferich primes reached an upper bound of 6.7e15, PrimeGrid's search will begin at 3e15. The reason for this is that Dorais and Klyve redefined what a "near" Wieferich prime was. Therefore, they did not search for "classical" "near" Wieferich primes. We do not expect to find any Wieferich primes between 3e15 and 6.7e15, but we do expect some "near" Wieferich primes. [EDIT - It appears that Dorais' and Klyve's definition did catch some "classical" "near" Wieferich primes but not all. We're searching for |A| < = 1000]

Classical Definition of nearness

From Wiki: "A prime p satisfying the congruence 2^((p−1)/2) ≡ ±1 + Ap (mod p^2) with small |A| is commonly called a near-Wieferich prime." Therefore, we are going to classify finds as follows:
Wieferich prime: A = 0
|A| <= 10
|A| <= 100
|A| <= 1000
Additional Information

Wieferich: The Prime Glossary
Wieferich: Math World
Wieferich: Wikipedia

For more in depth information, please see the following:


Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes", Math. Comp. 66 (217):433–449.
Dorais, F. G.; Klyve, D. (2011), "A Wieferich Prime Search Up to 6.7×10^15", Journal of Integer Sequences 14 (Article 11.9.2).
Knauer, Joshua; Richstein, Jörg (2005), "The continuing search for Wieferich primes", Math. Comp. 74 (251):1559–1563.
McIntosh, R. J. (2004), Wall-Sun-Sun (Fibonacci Wieferich) Search Status e-mail to Paul Zimmermann.

Welcome to the Wall-Sun-Sun Prime Search

A Wall–Sun–Sun (or Fibonacci–Wieferich) prime is a prime p > 5 in which p^2 divides the Fibonacci number , where the Legendre symbol is defined as

Although it has been conjectured that infinitely many exist, there are no known Wall–Sun–Sun primes. As of December 2011, if any exist, they must be > 9.7e14. We will begin our search at this limit.

The lack of success has lead to an interest in "Near" Wall-Sun_Sun primes. They are defined as special instances (with small |A|) of F_(p-(p/5)) (mod p^2). Here's a list of "Near" Wall-Sun_Sun primes (coming soon) with p > 200.

Search History

Search limit Author Year
1e9 Williams 1982
2^32 Montgomery 1991
1e14 Knauer and McIntosh 2003
2e14 McIntosh and Roettger 2005
9.7e14 Dorais and Klyve 2011

They are named after Donald Dines Wall and twin brothers Zhi-Hong Sun and Zhi-Wei Sun. Drawing on Wall's work, in 1992 the brothers proved that if the first case of Fermat's last theorem was false for a certain prime p, then that p would have to be a Wall–Sun–Sun prime.

Classical Definition of nearness

A prime p satisfying the congruence F_(p-(p/5)) ≡ Ap (mod p^2) with small |A| is commonly called a near-Wall-Sun-Sun prime. Therefore, we are going to classify finds as follows:
Wall-Sun-Sun prime: A = 0
|A| <= 10
|A| <= 100
|A| <= 1000
Additional Information

Wall-Sun-Sun: The Prime Glossary
Wall-Sun-Sun: Math World
Wall-Sun-Sun: Wikipedia
For more in depth information, please see the following:

McIntosh, R. J. (2004), Wall-Sun-Sun (Fibonacci Wieferich) Search Status e-mail to Paul Zimmermann.
Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes", Math. Comp. 66 (217):433–449.
McIntosh, R. J.; Roettger, E. L. (2007), "A search for Fibonacci-Wieferich and Wolstenholme primes", Math. Comp. 76 (260):2087–2094.
Dorais, F. G.; Klyve, D. (2011), "A Wieferich Prime Search Up to 6.7×10^15", Journal of Integer Sequences 14 (Article 11.9.2)..